Orthogonality

Orthogonality is a concept in mathematics describing perpendicularity in Euclidean geometry and, more generally, a relation between elements of an inner product space that are perpendicular. Two vectors u and v are orthogonal if their inner product ⟨u, v⟩ equals zero. In R^n with the standard dot product, this means u · v = 0. A set of nonzero vectors is orthogonal if every pair has inner product zero; it is orthonormal if, in addition, each vector has unit length.

Orthogonality is fundamental in simplifying projections and decompositions. The Gram–Schmidt process converts any finite set of linearly independent vectors into an orthogonal (or orthonormal) basis for its span. In function spaces with an appropriate inner product, such as ⟨f, g⟩ = ∫ f(x) g(x) w(x) dx, orthogonality means the integral is zero. Classical families of orthogonal functions include trigonometric functions in Fourier series and polynomials such as Legendre and Chebyshev polynomials. Orthonormal bases enable efficient representations of signals and functions and facilitate least-squares approximations.

Orthogonal complements: For a subspace W of a finite-dimensional inner product space, the set W⊥ of all vectors orthogonal to every vector in W is its orthogonal complement; dim(W) + dim(W⊥) = dim(V).

Applications span numerical linear algebra, signal processing, and statistics (where orthogonality implies uncorrelated components under certain assumptions). Orthogonality is defined relative to a chosen inner product, so different inner products yield different notions of orthogonality.

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